Title
Linearized topologies and deformation theory Linearized topologies and deformation theory
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Publication type
article
Publication
Amsterdam ,
Subject
Mathematics
Source (journal)
Topology and its applications. - Amsterdam
Volume/pages
200(2016) , p. 176-211
ISSN
0166-8641
ISI
000370897400012
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
Abstract
In this paper, for an underlying small category U endowed with a Grothendieck topology tau, and a linear category a which is graded over U in the sense of [13], we define a natural linear topology T-tau on a, which we call the linearized topology. Grothendieck categories in (non-commutative) algebraic geometry can often be realized as linear sheaf categories over linearized topologies. With the eye on deformation theory, it is important to obtain such realizations in which the linear category contains a restricted amount of algebraic information. We prove several results on the relation between refinement (eliminating both objects, and, more surprisingly, morphisms) of the non-linear underlying site (U, tau), and refinement of the linearized site (a, T-tau). These results apply to several incarnations of (quasi-coherent) sheaf categories, leading to a description of the infinitesimal deformation theory of these categories in the sense of [17] which is entirely controlled by the Gerstenhaber deformation theory of the small linear category a, and the Grothendieck topology tau on U. Our findings extend results from [17,12,7] and recover the examples from [21,20]. (C) 2015 Elsevier B.V. All rights reserved.
Full text (open access)
https://repository.uantwerpen.be/docman/irua/9c13f4/129685.pdf
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